Proof

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A proof is a mathematical argument used to verify the truth of a statement. This usually takes the form of an orderly series of statements based upon axioms. When a statement has been proven true, it is considered to be a theorem.

Proofs generally use an implication as the statement to prove. The goal of a proof is to show that for all values of a given number, object, etc., that if a given condition is met, the conclusion will be true. For example, the implication, "for all natural numbers n, if n is a prime greater than 2, then n is odd" gives the domain of the implication (n is a natural number), a condition or hypothesis (n is a prime greater than 2) and the conclusion (n is odd).

Mathematical induction

Mathematical induction seeks to show by implication that if a value is true for a given natural number, it is true for all natural numbers greater than that number. Induction is generally only applied to the natural numbers. The induction principle proceeds as follows:

Let P be a predicate with the natural numbers as its domain. Suppose that P has these two properties:

Complete induction

Complete induction is similar to mathematical induction, except that the hypothesis of the implication in the second property of implication is not only for P(n), but for all values less than or equal to n.

List of proofs

Etymology

The word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscis”, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),[1] and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.[2]

History

Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof.[3] It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement".[4] In particular, the ancient Egyptians had empirically discovered some truths of geometry, such as the formula for a truncated pyramid.[5] The Phoenician philosopher Thales (624–546 BCE) proved some theorems in geometry. In the Greek tradition of mathematics, Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. In HellenisticEgypt, Euclid (300 BCE) began with undefined terms and axioms (propositions regarding the undefined terms assumed to be self-evidently true, from the Greek “axios” meaning “something worthy”) and used these to prove theorems using deductive logic.

Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption that axioms are "true" in any sense; this allows for parallel mathematical theories built on alternate sets of axioms (see Axiomatic set theory and Non-Euclidean geometry for examples).

"Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer.

"Al-Karaji's argumen includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k − 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes."

"The central idea in ibn al-Haytham's proof of the sum formulas was the derivation of the equation [...] Naturally, he did not state this result in general form. He only stated it for particular integers, [...] but his proof for each of those k is by induction on n and is immediately generalizable to any value of k."